2080 JOURNAL OF PHYSICAL VOLUME 41

Lateral Mixing in the Pycnocline by Baroclinic Eddies

GUALTIERO BADIN Department of Earth Sciences, Boston University, Boston, Massachusetts

AMIT TANDON Department of Physics and Department of Estuarine and Sciences, University of Massachusetts—Dartmouth, North Dartmouth, Massachusetts

AMALA MAHADEVAN Department of Earth Sciences, Boston University, Boston, Massachusetts

(Manuscript received 10 January 2011, in final form 16 May 2011)

ABSTRACT

Using a process study model, the effect of mixed layer submesoscale instabilities on the lateral mixing of passive tracers in the pycnocline is explored. Mixed layer eddies that are generated from the baroclinic in- stability of a front within the mixed layer are found to penetrate into the pycnocline leading to an eddying flow field that acts to mix properties laterally along surfaces. The mixing of passive tracers released on such isopycnal surfaces is quantified by estimating the variance of the tracer distribution over time. The evolution of the tracer variance reveals that the flow undergoes three different turbulent regimes. The first regime, lasting about 3–4 days (about 5 inertial periods) exhibits near-diffusive behavior; dispersion of the tracer grows nearly linearly with time. In the second regime, which lasts for about 10 days (about 14 inertial periods), tracer dispersion exhibits exponential growth because of the integrated action of high strain rates created by the instabilities. In the third regime, tracer dispersion follows Richardson’s power law. The Nakamura effective diffusivity is used to study the role of individual dynamical filaments in lateral mixing. The filaments, which carry a high concentration of tracer, are characterized by the coincidence of large horizontal strain rate with large vertical vorticity. Within filaments, tracer is sheared without being dispersed, and consequently the effective diffusivity is small in filaments. While the filament centers act as barriers to transport, eddy fluxes are enhanced at the filament edges where gradients are large.

1. Introduction advection of freshwater create lateral gradients in prop- erties. Such horizontal gradients in the and In the oceanic mixed layer (ML), atmospheric forcing, can compensate each other in their contribution ocean dynamics, and their interplay act to leave the to density (Rudnick and Ferrari 1999) or give rise to ML surface waters well mixed. While the ML waters are density fronts. mixed in the vertical, lateral gradients in temperature ML fronts in the ocean are dynamically unstable. They and salinity are a common feature. Processes responsible evolve in response to geostrophic adjustment (Tandon for the creation of lateral gradients in temperature and and Garrett 1994, 1995; Young 1994) while undergoing salinity in the open ocean include nonhomogeneous heat baroclinic instability (Molemaker et al. 2005; Boccaletti and freshwater fluxes, wind mixing associated with the et al. 2007) and frontogenesis (Hoskins and Bretherton passage of a storm, and ocean convection. Near the coast, 1972; Capet et al. 2008a,b). A departure from quasi- the upwelling of deep water, tidal mixing, and estuarine geostrophic dynamics in the ML results in rich submeso- scale dynamics. Frontogenesis, which can be enhanced through interaction with the mesoscale strain field (Bishop Corresponding author address: Gualtiero Badin, Department of Earth Sciences, Boston University, 675 Commonwealth Ave., 1993; McWilliams et al. 2009a,b), is able to transfer energy Boston, MA 02215. from the mesoscale to the small scales where it is dissipated E-mail: [email protected] (Capet et al. 2008c). Fronts give rise to submesoscale

DOI: 10.1175/JPO-D-11-05.1

Ó 2011 American Meteorological Society NOVEMBER 2011 B A D I N E T A L . 2081

filaments of intense vertical vorticity with enhanced satellite imagery. We think of this process study as vertical velocities (Spall 1997; Mahadevan and Tandon examining lateral mixing during the growth phase of 2006), which are significant for the transfer of nutrients submesoscale ML instability, which could be viewed to the surface euphotic layer (Levy et al. 2001). The as growing on the edge of an underlying mesoscale ageostrophic secondary circulation that results in verti- feature. cal velocities can be further intensified by vertical mix- In what follows, we present the analysis of numerical ing (Nagai et al. 2006) and nonlinear Ekman effects experiments used to infer tracer mixing that is driven by (Thomas and Lee 2005). The presence of weak stratifi- submesoscale ML eddies generated from various frontal cation and large vertical shears in the ML leads to configurations. In section 2 we also review theoretical ageostrophic baroclinic instabilities (Stone 1966, 1970, results for tracer dispersion and particle statistics that 1971), which have much faster growth rates than baroclinic are useful for understanding tracer dispersion the sub- instability in the pycnocline (Boccaletti et al. 2007). De- mesoscale context. A reader familiar with these could spite being adiabatic, ML baroclinic instability results in skip directly to section 3, which describes our numerical the restratification of the ML (Fox-Kemper et al. 2008). experiments. Our results, in section 4a, start with a de- In contrast to earlier studies that have focused on the scription of submesoscale signatures in the pycnocline role of submesoscale dynamics on vertical advection and arising from surface ML instabilities. To study mixing, we mixing, as well as the rearrangement of buoyancy in the introduce a passive tracer along the initially flat isopycnal ML, the aim of this study is to infer the importance of surfaces at various depths below the ML. As the flow submesoscale dynamics on lateral mixing along isopycnal develops, we use the evolution of the dye distribution to surfaces in the oceanic pycnocline. This is motivated by infer the rate of lateral mixing. Our methods of analysis observations in which rates of lateral mixing were found to are based on previous observational studies where dye be time and scale dependent (Okubo 1970), and the in- was used to infer lateral mixing—for example, during the ferred horizontal diffusivity far exceeded the values that North Atlantic Tracer Release Experiment (NATRE) could be explained by shear dispersion (Sundermeyer and (Ledwell et al. 1998; Sundermeyer and Price 1998; Polzin Price 1998). What are the processes that lead to enhanced and Ferrari 2004) and on the New England continental lateral dispersion on scales of 0.1–10 km? Is such mixing shelf (Houghton 1997; Sundermeyer and Ledwell 2001; driven by dynamical instabilities and thereby associated Ledwell et al. 2004). Since the ML front and geostrophic with the growth and evolution of such instabilities? It has flow field are oriented west to east, and since our model been previously hypothesized that enhanced lateral mix- domain is zonally periodic, we lay down the dye in east– ing in the pycnocline results from the generation of mixed west, x-oriented streaks across the domain, and estimate patches generated by internal wave breaking that spin up dispersion in the cross-front, y direction. We then de- under the action of gravity and rotation to form vortical scribe the time evolution of the tracer concentration modes or eddies (Sundermeyer et al. 2005; Sundermeyer variance for streaks of tracer released on different iso- and Lelong 2005). In this study, we neglect vortical modes, pycnal surfaces in section 4b. We use the variance analysis and instead examine lateral mixing resulting from the to obtain an integrated measure of lateral mixing and as- submesoscale flow field generated by ML eddies. sess its dependence on ML dynamics. The mixing invoked A natural question is to ask whether the submesoscale by this mechanism is compared with values estimated for dynamics of the ML can impact the pycnocline. We ad- other processes in section 4c. In section 4d, we estimate the dress this question with a modeling process study of the effective diffusivity of an individual filament to mechanis- ML and pycnocline within a limited domain (approxi- tically understand the structure and role of submesoscale mately 200 km 3 100 km). Our three-dimensional nu- filaments in mixing. Finally, we present some points of merical ocean model is initialized with a zonally oriented discussion and our conclusions in sections 5 and 6. ML front, overlying an initially quiescent pycnocline with flat . The model domain is a zonally periodic channel. The ML is chosen to be deep (;200 m) so as to 2. Theoretical background generate a submesoscale flow field without any forcing. a. ML instabilities The model setup is described in section 3 and in the appendix. The numerical experiments capture the growth In the oceanic ML, ageostrophic baroclinic instability of submesoscale instabilities resulting in the spindown gives rise to dynamical features such as ML eddies of the front. In contrast, oceanic flows are highly developed and submesoscale filaments with typical length scales and most often forced by surface fluxes. Yet, submesoscale (Ls ; 1–10 km) that are much smaller than the Rossby instabilities are frequently triggered on the edges of meso- radius of deformation (Ld ; 10–100 km) associated with scale fronts and meanders as evidenced in high-resolution the pycnocline. Submesoscale dynamics are characterized 2082 JOURNAL OF VOLUME 41 by the Rossby (R0) and Richardson (Ri) numbers at- where H is the initial depth of the ML. It can thus be taining O(1) values locally, where R0 [ z/f is the ratio of seen that the submeso length scale associated with the 21 2 2 the vertical component of relative vorticity z 5 yx 2 uy to inertial time scale ;O( f ) is given by LS ; M H/f . 2 2 2 2 planetary vorticity f,andRi[ N /Uz ,whereUz 5 uz 1 While these scaling arguments apply to the ML, the 2 yz is the square of the vertical shear of the horizontal ve- theory can be extended to estimate the vertical extent of locity (u, y). The vertical and horizontal gradients of the the signatures into the pycnocline. Scaling (Pedlosky buoyancy b [2g[(r 2 r0)/r0], defined in terms of the 1987) suggests that the e-folding depth of penetration for density r and a reference value r0, give the square of an ML eddy is 2 the buoyancy frequency, N 5 bz and its horizontal equivalent, defined here only in terms of the cross-front fLs 2 D ’ , (5) direction as M 5 by. For an ML front aligned in the zonal N direction and in thermal wind balance, Ri ; O(1) gives rise to the balance (Tandon and Garrett 1994; Young where N, the buoyancy frequency in the pycnocline, is 1994; Tandon and Garrett 1995) assumed to be uniform. b. Turbulent dispersion of tracers 4 2 M N ; . (1) To examine the effect of submesoscale ML eddies on f 2 lateral mixing, we consider the evolution of a tracer with Such fronts develop large relative vorticity [z ; O(f)] concentration C(x, y, z, t) as described by the advection– and exhibit ageostrophic baroclinic instability, as stud- diffusion equation ied by Stone (1966, 1970, 1971). The growth rate of the ›C › ›C ageostrophic instabilities (Eldevik and Dysthe 2002) in 1 u $C 5 $ (k $ C) 1 k , (6) 2 ›t H H H ›z z ›z terms of Ri and the Burger number Bu 5 (Ls/L) , where 2 2 L is the width of the front and Ls 5 M H/f is the Rossby radius within the ML of depth H, is given by where kH is the horizontal diffusivity of the tracer, as- sumedtobeisotropic,andkz is the vertical diffusivity of the tracer. Because the slope of isopycnal surfaces in the 5/54 1/2 pycnocline is extremely small, it will suffice to consider the st 5 f . (2) 1 1 Ri 1 Ri Bu analysis in 2D (x, y), while projecting C and the horizontal velocity (u, y) onto the isopycnal surfaces. We focus on The wavelength for the linear growth of the most un- along-isopycnal (2D) dispersion of tracer because the ML stable perturbation is given by (Eldevik and Dysthe 2002) instabilities are adiabatic, particularly in the pycnocline. Calculations in which we included the vertical displace- 1 Bu 1/2 l 5 f L 1 1 1 , (3) ment of tracer showed no departure from our 2D results. 0 s Ri 2 Moments of the tracer concentration are used to study pffiffiffiffiffi the turbulent dispersion of the tracer along an isopycnal f 5 p ’ where 0 4 / 10 4. surface in response to the eddyingÐ flow (Thiffeault 2008). For Bu 5 0, (2) reduces to the formula for the growth Using the notation hi5 () dx dyjr, the zero moment rate found by Stone (1966), which compares well with the hCi defines the total amount of tracer, such that ›thCi 5 0 analytical solutions found by Eady (1949) and Fjørtoft for a conserved tracer. (1950) for Ri 1andRi5 0, respectively. It should be The zonal periodicity of our numerical experiments noted that ML instabilities act to modify both by and the lends itself to a simplification; we need consider only the ML depth H through restratification (Boccaletti et al. meridional (y directional) moments. The vector of first 2007; Fox-Kemper et al. 2008; Mahadevan et al. 2010), moments and tensor of second moments are, thus, both altering both Ri and Bu, and thereby the growth rate of reduced to just one component each. The first moment, the ML instabilities according to (2). normalized by the total tracer Given (1) and the thermal wind balance uz 52by/f, where u is the alongfront velocity, the cross-front eddy hyCi s 5 , (7) velocity y9 scales as (Haine and Marshall 1998; Fox- y hCi Kemper et al. 2008) defines the y displacement of the center of mass of the M2H y9;juj ; ju jH ; , (4) tracer. The second moment (normalized by the total z f tracer) minus the first moment squared, NOVEMBER 2011 B A D I N E T A L . 2083

2 2 hy2Ci of eddies then ky 1 and 1 2 J0 ’ (1/4)k y . When the s 5 2 s2, (8) yy hCi y separation scales larger than the eddy length scale then ky 1 and 1 2 J0 ’ 1. Assuming the power law de- 2a defines the tracer variance or dispersion in the y di- pendence E(k) } k , (10) can be separated as rection. Fickian diffusivity would cause tracer dispersion  ð ð (s ) to increase linearly with time as (›/›t)s ; K , d 2 1/y 1 ‘ yy yy y y 5 2 k2a k2y2 dk 1 2 k2adk, (11) where Ky is the diffusivity. dt 0 4 1/y c. Particle statistics which has solution Particle statistics offers a theoretical basis for un-  derstanding dispersion. Passive tracer dispersion statistics d 2 1 1 1/y 2 ‘ y 5 y2 k32a 1 k12a . (12) can be analyzed in terms of the statistical history of dt 2 3 2 a 0 1 2 a 1/y neutrally buoyant particles without inertia. For example, the method was used to study tracer dispersion by surface The first term of (12) diverges for a $ 3, while the sec- waves (Herterich and Hasselmann 1982). If particles are ond term diverges for a # 1. As noted by Bennett (1984) seeded in a patch of passive tracer, after an initial dif- and LaCasce (2008), for intermediate values of the fusion time the particles and the tracers will have the spectral slope 1 , a , 3, same domain of occupation (Garrett 1983). The disper- sion of a patch of particles is a function of the strain  d 2 rate g 5 [(u 2 y )2 1 (y 2 u )2]1/2. Initially, for time t , y } ya21, (13) x y x y dt O(g21) the patch grows diffusively (Garrett 1983). The initial period of small-scale diffusion lasts until the time so that that the tracer patch begins to be influenced by sub- mesoscale strain. For submesoscale dynamics, g 5 O(f) K } y(a11)/2. (14) (Mahadevan and Tandon 2006), so that for f 5 1024 s21 y the initial diffusion period for the tracer caught in a sub- mesoscale filament is only about 0.1 days. However, the The dependence of Ky on the particle separation thus di- tracer streak is caught by an ensemble of several in- rectly reflects the spectral slope a. In this case the disper- termittent filaments, resulting in a weaker ensemble strain, sion of particle pairs is dominated by eddies of the same which results in a longer diffusive period. size as the separation and the dispersion is termed ‘‘local.’’ The theory of dispersion statistics described in Bennett For a $ 3 the dispersion is dominated by eddies larger (1984) and LaCasce (2008) allows us to study the evolution than the separation scale and it is called ‘‘non local.’’ Given and transition of dispersive behavior in time. Taking two a certain spectral slope a, Eq. (12) and, consequently, (9) particles with initial separation in the meridional direction can be solved at different stages of evolution of the flow as follows. y0 at the initial time t0, the relative diffusivity of the par- ticles Ky at a later time t . t0 is defined as (Taylor 1921) 1) EARLY TIME ð t At early times, the particles are very close and the 1 d 2 Ky [ hy i 5 hy0yi 1 hy(t)y(t)i dt, (9) velocity difference between the particles is approximately 2 dt t 0 constant. The tracer experiences Brownian-like disper-

sion. Following Einstein (1905) and Taylor (1921), for Ky where y2(t) is the relative dispersion of the particles, y 5 constant, syy varies linearly in time as (d/dt)y is the pair separation velocity, and the link be- tween particle statistics and passive tracer is given by 2 syy 5 4Kyt. (15) y (t) ; syy(t). Following Bennett (1984), it is possible to express the mean square separation velocity as 2) INTERMEDIATE TIME  ð d 2 ‘ At intermediate times, the nearest particle pair ve- y 5 2 E(k)[1 2 J0(ky)] dk, (10) dt 0 locities are mutually correlated, but the particle sepa- ration scale is still much smaller than the length scale of where E(k) is the Eulerian kinetic energy wavenumber eddies. In this period, the KE spectra has k23 dependence spectrum and J0 is the first Bessel function. When the (Kraichnan and Montgomery 1980) and ky 1, so that particle separation scale is smaller than the length scale (12) and (9) give (Lin 1972) 2084 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

1/3 d. Effective diffusivity of tracers syy 5 exp(C1h t) and (16) Effective diffusivity (Nakamura 1996; Winters and 1/3 Ky 5 C1h syy, (17) D’Asaro 1996) is related to the complexity of the shape of the area occupied by a certain value of tracer concen- 2 where C1 is a nondimensional constant, h 5 nhhj$zj i is tration. Even though the effective diffusivity is calculated the enstrophy dissipation rate, and nh is the momentum as a function of tracer concentration, it is useful in char- viscosity. This period is also referred to as the enstrophy acterizing the role of specific spatial structures on mixing. inertial range. The exponential growth of relative dis- Effective diffusivity has been employed to study the mix- persion (16) would also be expected for 3D turbulence, ing of tracers by idealized Kelvin–Helmoltz billows but at much smaller scales than the scales resolved in our (Nakamura 1996; Winters and D’Asaro 1996)—the numerical experiment. stratospheric and tropospheric transport and mixing of tracers in different dynamical regimes. Chaotic advec- 3) LATER TIME tion and the role of stratospheric polar vortices as bar- When the particle separation reaches synoptic scales, riers to mixing tracers have been studied through both for 2D turbulence and ky 1 the KE spectra has k2(5/3) observations (Nakamura and Ma 1997; Haynes and dependence, so that (12) and (9) give (Richardson 1926; Shuckburgh 2000a,b) and modeled flows (Shuckburgh Batchelor 1950) and Haynes 2003). It has also been used for oceanic jets along the Antarctic Circumpolar Current that are 3 shown to act as barriers to mixing (Marshall et al. 2006; syy 5 C2t and (18) Shuckburgh et al. 2009). 1/3 1/3 2/3 Under the approximation that the evolution of a cer- K 5 C2 syy , (19) y tain tracer concentration class C takes place along iso- pycnals in the interior, (6) can be written as a diffusion where C is a nondimensional constant and 5 n h($u)2i is 2 h equation the energy dissipation rate. Equation (18) is referred to as Richardson’s power law, while (19) represents Richardson’s 4 2 ›C › 2 ›C /3 law for the particle separation y since syy ; y . This 5 L K (20) ›t ›A 0 eff ›A regime is also referred to as the energy inertial range. The presence of different turbulent regimes is contro- in area coordinates, where A(C, t) is the area of the versial, both in the context of the atmosphere and the tracer concentration class C and ocean. In the atmosphere, the exponential growth de- scribed by (16) has been observed by balloon experi- L2 ments launched near the tropopause in the Southern 5 k eq Keff H 2 (21) Hemisphere (Morel and Larcheveque 1974; Er-El and L0 Peskin 1981). Lacorata et al. (2004), however, making use of finite-scale Lyapunov exponents, reanalyzed the same is the effective diffusivity. In (20)–(21) we make the dataset to find that the growth followed Richardson’s law approximation that, along isopycnals in the interior, the (18). In the ocean, exponential growth has been observed along-isopycnal diffusivity is the same as the horizontal 2 in the Gulf Stream region by subsurface floats (LaCasce diffusivity kH. In (21), L0 is the area corresponding to and Bower 2000). However, Ollitrault et al. (2005) and the minimum length of the tracer concentration class C Lumpkin and Elipot (2010) found Richardson growth in contours, and the same region. Richardson’s law has been observed, for ð 1 › example, by Stommel (1949), while exponential growth 2 5 $ 2 Leq 2 ( C) dA (22) has also been observed by J. Price (1981, unpublished ›C ›A A(C,t) manuscript) though dye release experiments in the ML. ›A Further, a transition from exponential to Richardson- type growth has been observed by surface drifters in the is the area corresponding to the contours of the tracer Nordic Seas by Koszalka et al. (2009). concentration C, deformed by the action of the straining Finally, in the quasi-geostrophic limit, turbulence velocity field. For a careful derivation of (21) and (22) theory predicts that both active tracers (e.g., enstrophy) see Nakamura (1996) and Winters and D’Asaro (1996). and passive tracers are cascaded with horizontal to Various approximations can be adopted to obtain 2 2 vertical aspect ratio N/f (Kraichnan 1967; Haynes and a value for L0. For the stratosphere, L0 can be obtained Anglade 1997; Smith and Ferrari 2009). because of the zonal symmetry of the flows and can be NOVEMBER 2011 B A D I N E T A L . 2085 related to the slowest decaying mode of the diffusion TABLE 1. Model parameters. equation on a sphere (Shuckburgh and Haynes 2003). f Coriolis parameter 1024 s21 For the Southern Ocean, the zonal distribution of tracer 3 Lx Horizontal scale of the channel 96 3 10 m 2 3 allows a definition of L0 by applying a large horizontal Ly Meridional scale of the channel 192 3 10 m diffusivity to the domain in order to maintain zonal Htot Total depth 500 m 2 H Initial ML depth (reference integration) 200 m symmetry (Marshall et al. 2006). In the present study, L0 2 21 2 nh Horizontal eddy viscosity 5 m s is calculated as the initial value of Leq. However, it can 25 2 21 nz Vertical eddy viscosity 10 m s be noted that K is an integral measure: in a fully tur- 2 21 eff kH Horizontal diffusivity 5 m s 2 21 bulent flow, regions belonging to the same tracer con- kz Vertical diffusivity 5 m s 23 centration class can be detached from each other (i.e., Dr0 Initial density variation across front 0.2 kg m the flow changes its topology), and the deformation of (reference integration) b Maximal lateral buoyancy gradient at 20.9 3 1027 s22 a particular patch belonging to a certain tracer concen- y0 front (reference integration) tration class would result in assigning the same value of

Keff to other patches that are not experiencing defor- mation. To assign Keff 5 0 to regions not experiencing 2 deformation, Keff should be analyzed by dividing the stratification that has an initial maximum value of N 5 domains in smaller subdomains or—when the division 1.8 3 1024 s22 at 240 m and decreasing to N2 5 1025 s22 into subdomains is not possible because, for example, of at 500 m. Full details of the model are reported in the the dominance of advection—by paying extra care not appendix and in Table 1. A reference model integration to assign values of Keff to regions that are not deformed using an initial lateral buoyancy gradient by0 520.9 3 by the flow. Furthermore, the domain-integrated value 1027 s22 and ML depth H 5 200 m is the basis for the of Keff increases initially and then reaches a maximum. discussion in the next section, unless otherwise stated. The calculations presented in this study refer to the Several numerical experiments with variations to these period when Keff has reached a maximum and the tracer parameters were also performed, as summarized in Table 2. field has not already divided into patches. In the numerical integrations, the initial value of by ranges between 0.2by0 and 5by0, and H ranges between 100 and 200 m. Correspondingly, Ri and Bu range between Ri 5 3. Numerical experiments 22 23 4 3 10 and Bu 5 10 for the integration with 0.2by0, We perform a number of numerical simulations of and Ri 5 25 and Bu 5 4 3 1023 for the integration with a deep, geostrophically balanced ML front that overlies 5by0. a pycnocline, which is initially at rest with flat isopycnals. a. Initialization of tracer for studies of dispersion The model domain (of extent Lx 5 96 km, Ly 5 192 km, and depth 5 500 m) is a periodic zonal channel with To study the lateral dispersion of tracer in the pycno- solid boundaries in the meridional (y) direction and cline, east–west-oriented streaks of tracer are laid down a stratified interior overlying a flat bottom. The model beneath the ML front along isopycnal surfaces s 5 25.44, grid uses 1-km horizontal resolution and 32 grid cells in 26.19, 26.54, and 26.76, which are at initial depths ranging the vertical that increase in size over depth. The domain from 217 to 355 m (see appendix for details). The streaks is initialized with lighter water in the southern half of the are initially 1 km wide (corresponding to 1 grid cell) and ML and denser water to the north, with a vertical front 96 km in length in order to extend across the entire do- separating the two regions. The initial velocity field is main, and of a concentration of 1 tracer unit. Details of a geostrophically balanced frontal jet [u(y, z)] within the the initialization of the tracer streaks are reported in the

ML with vertical shear (uz 52by/f ) in thermal wind appendix and in Table 2. The dispersion of the tracer balance; the interior is initially quiescent with flat iso- (syy) is estimated along each of the isopycnal surfaces as pycnals. The ML front evolves without any active wind a function of time and provides an integrated measure of or buoyancy forcing, (e.g., representing the spindown tracer mixing by the submesoscale dynamics. of an ML front created by the passage of a storm), which To test the sensitivity of the results to the initial tracer leads to genesis of ML eddies that restratify the ML. A distribution, we performed experiments where the tracer three-dimensional, Boussinesq, free-surface, nonhy- streak width was given an initial Gaussian distribution over drostatic numerical model (Mahadevan et al. 1996a,b) 10 km (10 grid cells) and a maximum concentration of 1 is used with the same configuration as in Mahadevan unit. Also, the sensitivity of our results to the numerical et al. (2010). The model has constant horizontal mo- parameters in the model was tested by reducing both hor- 2 21 mentum viscosity nh 5 5m s and vertical viscosity izontal momentum viscosity and horizontal tracer diffu- 25 2 21 2 21 2 21 nz 5 10 m s . The pycnocline is flat, with nonuniform sivity to nh 5 1m s and kH 5 1m s . 2086 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

TABLE 2. Combinations of the initial cross-front buoyancy gra- eddy kinetic energy integrated over the ML. The result- dient b , ML depth H, horizontal viscosity n , horizontal diffusivity 21 y h ing growth period for ML instabilities is st ’ 0.4 days. In k H, and initial tracer streak width W0 used in numerical experi- comparison, (2) suggests a growth period for the ML ments. The reference integration is with H 5 200 m, by0 520.9 3 21 2 27 22 2 21 2 21 instabilities of s ’ 0.5 days, estimated with N 5 10 s , nh 5 5m s , kH 5 5m s , and W0 5 1 km (1 grid t 26 22 cell) in meridional extent. Tracer was introduced on four isopycnal 10 s ,Ri5 1, and Bu 5 0.0023, where Bu is calcu- surfaces s 5 25.44, 26.19, 26.54, and 26.76 in the pycnocline. For the lated using L 5 32 km. Similarly, (3) gives a wavelength reference integration, the four isopycnal surfaces have an initial for the linear growth of the most unstable perturbation depth of 216.59, 256.88, 302.72, and 354.87 m, respectively. The for the ML eddies of l 5 1.1 3 104 m. 10-km-wide tracer streak has a Gaussian profile in meridional sec- tion with maximum value 1, whereas the 1-km-wide streak is simply The ML eddies that develop from the instability of the initialized with concentration 1 within the grid cell. front are seen to have a strong submesoscale character. The vertical component of the relative vorticity z and the Hb n k W y h H 0 lateral strain rate g (Figs. 1d–i) are elevated in filaments 2 21 2 21 200 m 0.2by0 0.5by0 0.75by0 by0 5by0 5m s 5m s 1km where they attain values of O(f). Notably, the elevated 2 21 2 21 150 m — — — by0 —5ms 5m s 1km 2 21 2 21 values of z and g occur at the same location in filaments, 100 m — — — by0 —5ms 5m s 1km 2 21 2 21 undermining the physical basis for a separation of sub- 200 m — — — by0 —1ms 1m s 1km 2 21 2 21 200 m — — — — 5by0 1m s 1m s 1km mesoscale flow into strain- and vorticity-dominated re- 2 21 2 21 200 m — — — by0 —5ms 5m s 10 km gions by the Okubo–Weiss parameter (Okubo 1970; 2 21 2 21 200 m — — — by0 —1ms 1m s 10 km Weiss 1991). Vertical sections across the front, shown at various times of the simulation in Fig. 2, reveal that ML eddies b. Tracer initialization for estimation of effective slump the isopycnals (shown in black) at the front, diffusivity causing restratification within the ML as they release the available potential energy associated with the lateral To estimate an effective diffusivity (K )weinitialize eff buoyancy gradient at the front. The vertical component a second tracer on the same isopycnals where the tracer of the relative vorticity z (shown in color) reveals that streaks were laid down with a continuous distribution ML eddies are often, but not always, surface intensified; C(y) 5 1 2 y/L ,whereL is the meridional dimension of y y yet, they are able to penetrate into the pycnocline to the domain and 0 # y # L . Because K depends on y eff result in the significant perturbation of isopycnal sur- tracer concentration classes, this tracer, which varies faces and an energetic adiabatic flow field that leads to linearly in the zonal direction from 1 to 0, allows us to vigorous lateral mixing of tracers. The evolution of the visualize K in the entire domain. During the numerical eff tracer streaks on two of the isopycnal surfaces as a result integration, the maximum concentration of tracer de- of such mixing is shown in cross section in Figs. 2d–f. creases exponentially, thus requiring a large number of Figure 3 shows the effect of the ML eddies on rela- classes of tracer concentration in order to resolve K eff tively shallow and deep isopycnal surfaces, s 5 25.44 even when the maximum concentration of tracer reaches (Figs. 3a–c) and 26.76 (Figs. 3d–f), 20 days after the very small values. We use 50 classes of tracer concen- spindown of the front begins. Along s 5 25.44, the ML tration, each having a range of 1/50. A similar analysis with eddies induce large deformations (630 m) in the ini- a larger number of tracer classes showed convergence, tially flat isopycnal surface. Filaments with jzj ’ 0.5f and suggesting that 50 tracer classes are adequate. g ’ f are visible at the same location in regions with strong gradients in the anomalous isopycnal depth. 4. Results Along s 5 26.76, the ML eddies deform the isopycnal (610 m) and create filaments with jzj ’ 0.5f and g ’ a. Growth of submesoscale instabilities 0.5f. The positions of the dynamic filaments demarcated The ML front, which is initially oriented east–west, by elevated z and g are not in the same location on the becomes unstable, forming meanders that are initially of two isopycnals, but are vertically deflected by the strati- a small wavelength, but grow as the instability develops fication. The vertical deflection of the filaments by strat- and form eddies that act to restratify the ML, moving ification and the development of thin horizontal layers lighter water over denser water with minimal vertical of independent vortex dipoles (Fig. 3c) is possibly anal- mixing. Figures 1a–c show the evolution of the surface ogous to experimentally observed zigzag instabilities density field in the numerical model. (Billant and Chomaz 2000a,b), which can grow in the The initial linear growth rate of the ageostrophic in- presence of both a low horizontal Froude number and stabilities in the model can be calculated in the linear a high Reynolds number, which are present in our model growth regime as st 5 log[EKE(t)]/2t, where EKE is the simulation. NOVEMBER 2011 B A D I N E T A L . 2087

FIG. 1. The time evolution of ML instabilities is shown in terms of (a–c) surface density (sigma), (d–f) vertical component of the relative vorticity (z), and (g–i) horizontal strain rate g, all plotted for the surface on days (top) 10, (middle) 15, and (bottom) 20 of the simulation. Here z and g are normalized by f. The domain is 96 km (in x) 3 192 km (in y); only the frontal region between y 5 50 and 150 km is shown.

The e-folding depth of penetration for the ML eddies, restratifying action of the ML eddies. Further work is 2 according to (5), is D ’ fLs/N ’ (10 m). The vertical required to explain the time evolution of the vertical structure of the ML eddies is, however, variable in time, structure of the instabilities. with features progressively penetrating deeper into the b. Lateral dispersion of tracer pycnocline (Fig. 3c) despite the nonuniform stratifica- tion that acts to restrict the barotropization of the flow, To study the along-isopycnal mixing of tracers in the at least in the quasi-geostrophic approximation (Smith pycnocline, we introduced tracer streaks beneath the ML and Vallis 2001). Theories to predict the correct vertical front on isopycnal surfaces at varying depths as shown in structure of the flow, such as surface quasi-geostrophy Fig. 2. The time evolution of the tracer concentration (SQG; e.g., Held et al. 1995; LaCasce and Mahadevan along s 5 25.44 (Figs. 4a–c) shows that, under the action 2006) are based on N2 being constant; here N2 is not of ML instabilities, the tracer streak begins to form wiggles uniform in the vertical and varies in time because of the (asseenonday10)thatgetstretchedinthecross-front 2088 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

FIG. 2. (a–c) Vertical (y–z) sections of density (continuous bold lines) and relative vorticity z normalized by f (colors) along x 5 20 km, and (d–f) vertical sections of density (continuous bold lines) and tracer concentration. Results are shown up to a depth of 400 m on days (top) 10, (middle) 15, and (bottom) 20 of the simulation. direction. On day 15, the regions of high tracer con- tracer filament slope: (DC/Dz)/(DC/Dy) 5 102. Taking centration are linked to each other in a filament that is a typical value of the buoyancy frequency in the interior folding on itself (Fig. 4b). On day 20, we find that the of N2 5 1024 s22, it is possible to see that [(DC/Dz)/(DC/ previously continuous filament is broken into discrete Dy)]2 ; N2/f 2 5 104, which is in agreement with the blobs of tracer that evolve independently of each other results of Haynes and Anglade (1997) and Smith and (Fig. 4c). The time evolution of the maximum tracer Ferrari (2009) for tracer filaments in the interior subject concentration (Fig. 4a, insert) shows that for the first 3–5 to both horizontal strain and vertical shear. Though we days, the maximum tracer concentration does not change. do not explore the role of vertical mixing in this study, Between days 5 and 15, the maximum tracer concentra- Haynes and Anglade (1997) and Smith and Ferrari tion decays rapidly and exponentially, following which, (2009) have shown that on account of the strong vertical the decay rate slows considerably. At greater depths, shear that tilts filaments, even a small vertical diffusivity along s 5 26.76, the tracer undergoes a similar evolution is sufficient to arrest the lateral downscale cascade of (Figs. 4d–f), but with a delayed response to the instabilities. tracer variance. At depth, the maximum tracer concentration decays at The analysis of the tracer displacement from the center a slower rate and asymptotes toward a larger value than of mass shows that along s 5 25.44, the distribution is was reached on s 5 25.44. initially Gaussian. At later times, the tracer displacement Vertical sections of the tracer streaks taken along x 5 shows significant deviation from the normal distribution 20 km show that the tracer is mainly spreading along (Figs. 5a–c). Analysis of the tracer displacement at the isopycnals (Figs. 2d–f). By day 20, the streak has spread same time, but at greater depth [e.g., along s 5 26.76 (Fig. from the initial position approximately 10 km in the 5d)] and for the integrations with a decreased value of the horizontal direction and 100 m in the vertical direction. cross-front density contrast 0.2by0 (Fig. 5e) and initial ML The ratio of the distribution of the tracer in the vertical depth H 5 100 m (Fig. 5f), also show deviations from the to the horizontal direction gives a typical value of the Gaussian distribution, even if much less pronounced. NOVEMBER 2011 B A D I N E T A L . 2089

FIG. 3. Dynamical properties on day 20 of the simulation shown on a shallow and deep isopycnal surface in the pycnocline. (a) Depth anomaly (m), (b) relative vorticity z normalized by f, (c) lateral strain rate g normalized by f along the isopycnal s 5 25.44; (d) depth anomaly (m), (e) z/f, and (f) g/f along the isopycnal s 5 26.76. Black line at x 5 20 km indicates the location of the vertical section in Fig. 2. 2090 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

FIG. 4. Evolution of tracer concentration along (left) s 5 25.44 and (right) 26.76 for days (a),(d) 10, (b),(e) 15, and (c),(f) 20. Embedded in (a) and (d): time evolution of the maximum tracer concentration along (a) s 5 25.44 and (b) 26.76.

To analyze the dispersion of tracer by the eddying gets caught in dynamic filaments with high strain rates, flow in the pycnocline, we estimate the tracer variance and gets rapidly stretched in the cross-front direction syy, defined in (8), as a function of time. We find, as will leading to an exponential growth in the variance. Con- be shown in the following analysis, that only at early sequently, a stage is reached when the tracer filaments times does the tracer behave diffusively. Thereafter, it break up, and the variance shows a power law behavior NOVEMBER 2011 B A D I N E T A L . 2091 with time. A Fickian description for the tracer’s be- ML generates stronger filaments and larger hgi (Fig. 6c) havior is thus valid only at early times. as expected based on the scaling for submesoscale flows A log–log plot of the time evolution of the tracer in (4). Consequently, this case results in a higher growth variance syy (Fig. 6a) clearly shows the existence of rate of tracer variance (Fig. 6d). Similarly, because the three different dynamical regimes, as described below. EKE and strain rate decays with depth, the upper pyc- nocline has larger values of hgi and shows higher growth 1) DIFFUSIVE REGIME rates. During the first 3–4 days, corresponding to about 6 3) RICHARDSON REGIME inertial periods, the instabilities grow linearly and tracer dispersion increases linearly in time. In this period, the From days 16 to 42 the tracer concentration variance 1.1 2.5 tracer variance evolves as syy ; t (Fig. 6a). Around along the isopycnal evolves as syy ; t (Fig. 6a). In this day 4, the evolution of tracer variance undergoes a sud- period, the flow has attained finite amplitude in- den transition into the second regime. stabilities. The tracer streak is broken into tracer blobs that evolve independently (Fig. 4c). 2) EXPONENTIAL REGIME The time evolution of the tracer variance suggests the From days 4 to 15, corresponding to 14 inertial periods, presence of two-dimensional turbulence in the inverse the tracer exhibits superdiffusive behavior, and the con- energy cascade range with 23 slope in the KE spectrum. 6.8 centration variance grows as syy ; t (Fig. 6a). In this The power law dependence of tracer variance in the period, tracer filaments are seen to show large excursions third regime is in agreement with upper-ML observa- 2.3 in y (Fig. 4b) and the growth of tracer filaments occurs at tions by (Okubo 1971), which reported syy ; t . The an exponential rate proportional to their strain rate (g). To transition time between the second and the third regime explain the rapid rate of increase in tracer variance by is in agreement with the observations of surface floats by exponential growth of tracer filaments, we plot the time Koszalka et al. (2009). evolution of the tracer variance in semilogarithmic space Analysis of the time evolution of the KE spectra (Fig. 6b). The exponential growth rate in this second, (Fig. 7a) shows the initial excitation of the kinetic energy transient, regime suggests the presence of two-dimensional on the ’10 km scale and the subsequent nonlinear turbulence in the enstrophy cascade range (Lin 1972). Such cascade to larger scale. Analysis of the KE spectrum at exponential growth has been observed in the oceanic ML day41(Fig.7b)showsthepresenceofk25/3 slope at syn- using tracers (J. Price 1981, unpublished manuscript) and optic scales and k23 slope at smaller scales, confirming surface drifters (Koszalka et al. 2009), and, subsurface of the presence of an inverse energy cascade range and an the Gulf Stream, using floats (LaCasce and Bower 2000). enstrophy cascade range. At even smaller scales, the spectra Following Garrett (1983), (16) can be modified to in- become steeper because of the action of viscosity. The clude the effect of strain acting on the tracer streak as slope of the spectra is the same along different isopycnals follows: at different depths.  To test the sensitivity of the results to the initial tracer Ky 1 distribution, we performed experiments where the tracer s 5 p exp C hgi t 2 . (23) yy hgi 3 4hgi streak width was given an initial Gaussian distribution over 10 km (10 grid cells) and a maximum concentration Here, hgi is the ensemble strain rate acting on the tracer of 1 unit. Results do not differ from those obtained with streak and C3 is a constant of O(1). While g can be dif- the 1-km-wide streak; we see the presence of the three ficult to control because of the intermittency of the strain regimes and the first regime shows the same time de- rate associated with filaments in both space and time, it is pendence in syy. Also, the sensitivity of our results to the possible to create different numerical realizations that numerical parameters in the model was tested by re- change hgi through the deployment of tracers at different ducing both horizontal momentum viscosity and hori- 2 21 depths inside the pycnocline, or through a change of the zontal tracer diffusivity to nh 5 1m s and kH 5 initial parameters of the ML front as shown in Fig. 6c. We 1m2 s21. Again, for the first regime, the slope of the compare results from the upper and lower pycnocline for tracer variance is the same as the value found for the the reference integration (full and dashed lines in Figs. larger values of horizontal diffusivity and viscosity, im- 6c,d), and in the upper pycnocline for a numerical in- plying that our results do not depend on the specific tegration with initial ML depth of 100 m. Figure 6d choice of these model parameters. shows that the exponential growth rate (slope in semi- It is of interest to estimate a lateral diffusivity for the logarithmic space) of the tracer variance for each of these tracer as a measure of mixing, but, since the tracer var- cases varies with hgi in accordance with (23). The deeper iance shows non-Fickian behavior in this regime, it 2092 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

IG F . 5. Normalized histogramsq offfiffiffiffiffiffiffiffiffiffi quantity of tracer in function of the meridional displacement (m) of the tracer from its center of mass, (y 2 y0) 2 syy, where y0 is the meridional position of the center of the domain. (left) Results from the reference integration along s 5 25.44 for days (a) 10, (b) 15, and (c) 40, representing the first, second, and third regime, respectively. (right) We show (d) the reference integration at day 40, along s 5 26.76; (e) the integration

with by0 diminished to 0.2by0 at day 60; and (f) the integration with initial ML depth decreased to H 5 100 m at day 60. Dashed lines represent the Gaussian distribution with standard deviation calculated from the tracer variance. Non- Fickian dispersion in the second and third regime leads to a deviation from the Gaussian distribution that is expected of a diffusive process. The non-Fickian behavior sets in at later times on deeper isopycnals and when the sub- mesoscale ML eddies are weaker (i.e., for smaller H and by). NOVEMBER 2011 B A D I N E T A L . 2093

FIG. 6. Time evolution of the tracer concentration variance syy along s 5 25.44 plotted on (a) logarithmic and (b) semilogarithmic scales. (c) Time evolution of the area-averaged strain rate (normalized by f ) hg/f i on different isopycnals (shallow and deep) and with ML H 5 100 m (for the shallow isopycnal). The evolution of the strain rate also exhibits a transition from one regime to another, but remains consistently higher for the shallower isopycnal and the simulation with deeper mixed layer. (d) Time evolution of the tracer concentration variance along s 5 25.44 (bold line) and along s 5 26.76 (dashed line) for the reference integration, and along s 5 25.44 for the numerical integration with initial ML depth of 100 m (dotted-dashed line). In (a), we find three distinct regimes characterized 2 1:1 6:8 2:5 by sy ; t , t , and t . In (b) and (d) we find that the second regime is near exponential. Thin straight lines rep- resent linear fits to estimate the exponential growth rate in the second dynamical regime. would not be appropriate to assume a constant lateral This definition of L isbasedonthefactthatifthetracerhas diffusivity for the tracer according to (15). Instead, we a Gaussian distribution in the meridional direction, most of qffiffiffiffiffiffiffiffiffi examine the time-dependent, apparent diffusivity (Ka), the tracer will be contained within 3 syy.Thetimeevo- expressed in accordance with Richardson (1926) and lution of Ka and L (Fig. 8a) shows that both quantities co- Okubo (1971) as vary in time, and both exhibit the three dynamical regimes noted for s . However, initially, the dispersive length scale s qffiffiffiffiffiffiffiffiffi yy K (t) 5 yy . (24) 3 s is smaller than what can be resolved by the model, a 4t yy and L and Ka are meaningful only at later times (i.e., in the third regime). Plotting Ka versus L for the third regime For Fickian diffusion, Ka would be constant and equiv- (Fig. 8b) reveals that the modeled tracer shows the behavior alent to Ky in (15) (Einstein 1905; Taylor 1921). Using a wide range of observations, Okubo (1971) showed that K ;L1:24. (26) Ka varies with the diffusive length scale (L) defined (for a dispersion in the y direction) as This 1.24 power dependence inferred for the model pffiffiffiffiffiffiffi simulations is higher than the value of 1.1 obtained from L53 s . (25) yy observations by Okubo (1971), but lower than the 2094 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

FIG. 7. (a) Time evolution of the kinetic energy spectra along s 5 26.76 for the reference integration. (b) Kinetic energy spectra at day 41 for the reference integration along s 5 25.44 (full bold line), 26.19 (full thin line), 26.54 (dashed line), and 26.76 (dotted-dashed line).

4/3 Richardson power law in (19) that is proposed to hold in the ocean (Stommel 1949; Ollitrault et al. 2005; Lumpkin and Elipot 2010). We make a further effort to estimate a time-dependent horizontal diffusivity Ky as a measure of lateral mixing by using each of the three proposed relationships (15), (17), and (19) within each of the three regimes. To do this, we must first estimate the transition times between the diffu- sive and exponential regimes and between the exponential and Richardson regime. In particular, the transition time between the diffusive and exponential regimes can be FIG. 8. (a) Evolution in time of the apparent diffusivity Ka (m2 s21) (full line) and diffusive length L (m) (dashed line). (b) calculated from (15) to (17) as Plot of the apparent diffusivity vs the diffusive length for the third 2 21 regime. (c) Evolution in time of the lateral diffusivity Ky (m s ) 1 1 along s 5 25.44 (full line) and 26.76 (dashed line) for the reference t 5 , (27) 1,2 1/3 integration, along s 5 25.44 for the integration with initial by set as 4 C1h 0.2by0 (dotted-dashed line), and along s 5 25.44 for the integration with initial ML depth H 5 100 m (dotted line). NOVEMBER 2011 B A D I N E T A L . 2095 where the constant C1 can be found from linear regression from the reference integration. The transition time be- tween the exponential and Richardson regime (t2,3)hasno simple analytical form and is thus estimated from the transitionintherateofincreaseinvariance.

We then calculate Ky in each of the three regimes using (15), (17), and (19), respectively, having estimated h and —the enstrophy and energy dissipation rates. The re- sulting Ky is shown in Fig. 8c for different model simu- lations. For the reference integration (full line), during 22 2 21 the first regime Ky has the value of 10 m s . During the second regime Ky increases to approximately 2300 m2 s21. Finally, during the third regime, the growth rate of Ky decreases. By the end of the integration, Ky 2 21 reaches the value of approximately 2500 m s .Wesee 2 FIG. 9. Evolution in time of the tracer variance syy (m ) along a strong dependence of Ky on the depth of the isopycnal s 5 25.44 for tracer deployed in a streak at the beginning of the along which the tracer is released (dashed line), the lat- numerical integration (full line) at days 10 (dashed line) and 20 eral buoyancy gradient in the ML by (dotted-dashed line), (dotted-dashed line). and on the initial ML depth, H (dotted line). The exact scaling dependence is, however, not clear. In general, results can be influenced by the relationship between idealized model for tracer dispersion in the ocean by the the depth of the isopycnal on which Ky is calculated and internal wave field. Using the parameters utilized in our the value of N2 in the pycnocline. In fact, changes of the study, the single-particle diffusivity [Holmes-Cerfon et al. initial ML depth H also result in changes in N2, and these 2011, their Eq. (8.6)] leads to a horizontal diffusivity can affect the penetration depth of the ML eddies and for the Garrett–Munk (e.g., Munk 1981) internal wave 2 21 then can affect the lateral mixing. spectra of Ky ’ 0.12 m s , which is an order of Analysis of the tracer variance at different depths and magnitude larger than the diffusivity induced by ML for different initial values H and by (given in Table 2) eddies in the pycnocline during the first regime. The show that the various simulations differ in the rate of diffusivity found by Holmes-Cerfon et al. (2011) is, growth of tracer variance. But, the presence of the three however, limited by the fact that it is based on one- turbulent regimes described in section 4b, and the 1.24 particle dispersion. Multiple-particle dispersion is ex- power law between apparent diffusivity (Ka) and the pected to give larger values for the interior horizontal diffusion scale (L) are robust for all the cases. diffusivity induced by internal waves. Our values for Ky Since these numerical experiments were started from an are also compared with the horizontal diffusivity due to idealized initialization of a front, one may ask how these internal wave shear dispersion. Using the relationship analyses might apply when tracer is introduced in a flow found by Young et al. (1982) for internal wave shear field that has already spun up, as when dye is injected in dispersion with parameter values used in this study, we 2 21 the ocean. To address this question, we injected the tracer estimate Ky ’ 0.04 m s ,whichisafactorofthree streaks at 10 and 20 days after model initialization. Our lower than the single-particle dispersion estimate results, shown in Fig. 9, reveal that the tracer variance Holmes-Cerfon et al. (2011), but comparable or larger grows very abruptly when the tracer is introduced, so that than our estimate for interior diffusivity by ML eddies within a short period it catches up to the level that it would during the first regime. Finally, our values for Ky can have attained had the tracer been introduced at the be- also be compared with the horizontal diffusivity due to ginning of the model integration. vortical modes generated from the adjustment of mixed patches following diapycnal mixing events. Using the c. Comparison with other mechanisms for horizontal relationship for K in Sundermeyer and Lelong (2005), mixing y their Eq. (9), along with parameter values used in our 2 21 The values found for Ky can be compared with the study gives Ky ’ 2m s , which is larger than internal values found from other mechanisms for horizontal wave-driven mixing and the value of interior diffusivity mixing acting in the oceanic pycnocline. Holmes-Cerfon generated by ML eddies in the first regime within this et al. (2011) calculated the horizontal particle dispersion study. At early times in the development of ML eddies due to random waves in a three-dimensional rotating (i.e., within the first regime), it would be difficult to and stratified Boussinesq system, which acts as an distinguish the effect of ML eddies on mixing in the 2096 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41 pycnocline from other mechanisms. But, when devel- j[z^3 u] $uj a 5 , (28) oped (i.e., in regimes 2 or 3), ML eddies provide a dis- juj tinct enhancement in mixing, far beyond what can be achieved by other mechanisms like internal waves and to be enhanced in the core of the filament. However, vortical modes. shear flows also exhibit diffusion in the form of Taylor d. Effective diffusivity of filaments dispersion (Taylor 1953), which acts along the tracer concentration contours rather than across them. To better understand the role of singular submesoscale For an idealized 2D tracer filament balanced by the filaments in lateral mixing, we use a tracer that was ini- action of horizontal strain g in the x direction and hor- tialized on an isopycnal surface with a uniform gradient in izontal diffusivity kH, the advection–diffusion equation y. To isolate the effect of submesoscale filaments, we use can be written as the simulation with a fivefold stronger lateral buoyancy gradient (b 5 5b ), which produces pronounced fila- ›C ›C ›C ›2C ›2C y y0 1 gx 2 gy 5 k 1 . (29) ments. Figure 10a shows the tracer concentration on day ›t ›x ›y H ›x2 ›y2 18 of the integration. We focus, hereafter, on a sub- domain (Fig. 10b) containing a distinct filament that ex- Assuming a balance between the along-filament strain hibits high relative vorticity (z; Fig. 10c), strain rate (g; and the across-filament diffusivity, (29) predicts a typical Fig. 10d), and potential vorticity (Fig. 10e). Both z and g filament width (Garrett 1983) exceed f, highlighting the submesoscale character rffiffiffiffiffiffiffi (Mahadevan and Tandon 2006) of the filament. k W ; H. (30) The quantity Keff is estimated using (21) and (22) and g shown in Fig. 11a. Surprisingly, the filament, which is in- 2 21 strumental in rapidly enhancing the dispersion of tracer, For kH 5 5m s and g 5 O(f), (30) yields W ’ 200 m. is delineated by a region of zero Keff at the filament center, However, the strong horizontal shear implies that Taylor 2 21 with Keff ’ 600 m s on the edges. Its effect on tracer dispersion can also contribute to filament lengthening. fluxes becomes apparent when we examine the lateral 2 2 Assuming uH 5 gx 1 U0[(1 2 y /W )]x^,wherey is now tracer flux Keff $C in Fig. 11b. Enhanced fluxes are found the cross-filament coordinate with respect to the filament along the filament boundary. To test the dependence of center about which the flow is assumed axisymmetric and the barrier behavior of filaments on numerical param- U0 is a typical along-filament velocity, after a Reynolds 2 21 eters, the model was integrated with nh 5 1m s and decomposition, (29) yields (Taylor 1953) 2 21 kH 5 1m s . Results (not shown) show the presence of "# filaments with zero Keff at the filament center and with ›C ›C ›C (gx)2 ›2C 1 gx 2 gy 5 k 1 1 mW2 enhanced fluxes are found along the filament boundary, ›t ›x ›y H k2 ›x2 thus suggesting the independence of the result to model H parameters. ›2C 1 k , (31) The region of zero effective diffusivity in the core of H ›y2 the filament suggests that the filament is a barrier to mixing in its interior and must therefore carry tracer where m is a nondimensional constant of order 1. The without mixing it in its core, but eventually exchange it Taylor dispersion term enhances the effect of the hor- with the surrounding region across its edges. This izontal strain in elongating the filament. Assuming 2 Lf 2 2 2 behavior is different from that of mixing barriers as- W [(g ) /kH]W 1, where Lf is the typical length of 2 21 sociated with open-ocean jets, where mixing is sup- the filament, for kH 5 5m s and g 5 O(f) the Taylor pressed by the mean flow (Marshall et al. 2006; Smith dispersion and the horizontal diffusivity terms are both and Marshall 2009; Abernathey et al. 2010; Ferrari of O(1024 C). The balance between Taylor dispersion 2 2 2 and Nikurashin 2010). In submesoscale filaments, the and the horizontal diffusivity gives W g /kH ; kH/W , mechanism generating zero Keff in the filament is hori- which results in the same balance as (30). Along-filament zontal shear, which acts to move the tracer contours Taylor dispersion resulting from the large lateral shear in parallel one to each other without deforming them. The filaments can thus have a similar contribution as the lat- hypothesis that the center of the filament is dominated eral strain rate in elongating tracer filaments. Its neglect by shear flow is supported by the observation that it could thus lead to an overestimation in the width of the contains both high-vorticity and high-strain rate at the tracer filaments. same location. Figure 10f shows the horizontal shear, The order of magnitude of Keff is in agreement with the calculated as relationship between the Nusselt number Nu 5 Keff/kH NOVEMBER 2011 B A D I N E T A L . 2097

FIG. 10. (a) Tracer concentration for the entire domain along s 5 25.44 at day 18 for the 5by0 integration. The following focus on a subregion: (b) tracer concentration, (c) relative vorticity (z) normalized by f, (d) lateral strain rate (g) normalized by f, (e) potential vorticity (1029 m21 s21), and (f) horizontal shear normalized by f for a sub- mesoscale filament forming in the pycnocline.

and the Peclet number Pe 5 UL/kH of the flow independent of kH (Shuckburgh and Haynes 2003; (Shuckburgh and Haynes 2003; Marshall et al. 2006). For Marshall et al. 2006). a filament (Fig. 10a) with typical velocity of O(1 m s21) and typical cross-filament size of O(10 km), Pe 5 2 2 5. Discussion O(10 ), which corresponds to Nu 5 Keff/kH 5 O(10 ). The value of Pe found through the simple scaling Previous work has examined submesoscale instabilities suggests that the flow is in a regime in which Keff is within the mixed layer and their role in the vertical 2098 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

experiments, the action of ML eddies results in a non- constant lateral diffusivity in the pycnocline that cannot be represented by Fickian diffusion, except at early times. Filaments that are generated by the mesoscale strain field have been known to have an effect on tracers. In a 2D mesoscale flow field, the cross-filament diffusion of tracer has been conjectured to arrest the elongation of the fila- ment due to strain (Garrett 1983). In 3D, vertical diffusion comes into play on account of the strong vertical shear, and arrests filamentation (Haynes 2001; Smith and Ferrari 2009). Here, we examine submesoscale filaments, in which both the horizontal shear and strain rate are of the same order. For such filaments, Taylor dispersion arising from horizontal shear could enhance the growth of the filament. Our scaling analysis suggests that it may be important to consider the horizontal shear, in addition to the lateral strain rate, in estimating the cross-filament diffusivity. Finally, the processes described here are likely to play a role in the distribution and fluxes of biogeochemical properties in the upper ocean. Even though the bio- geochemical sources and sinks for nitrate are patchy, the correlation between density and nutrient is generally very robust in the pycnocline (e.g., Kamykowski and Zentara 1986). The dynamics described here may be of importance in homogenizing properties, such as nutri- ents, along isopycnals.

6. Conclusions A series of numerical simulations were conducted to isolate the effect of ML eddies on the lateral mixing of tracers in the ocean interior. Our results show the im- portance of ML processes in influencing the dynamics 2 21 FIG. 11. (a) Effective diffusivity (m s ) and (b) lateral tracer and the mixing in the pycnocline. The strength of along- 21 flux along isotracer lines (m s ) for a submesoscale filament isopycnal lateral mixing is highly variable and de- forming in the pycnocline along s 5 25.44 at day 18 for the 5by0 integration. pendent on parameters like the ML depth and lateral buoyancy gradient in the ML, as well as the isopycnal depth and vertical buoyancy gradient within the pyc- exchange of properties between the surface and pycno- nocline. The evolution of tracer streaks injected in the cline. Lateral mixing in the pycnocline had been hypoth- pycnocline shows the presence of three distinct turbu- esized to result from interior processes like vortical modes lent regimes characterized by differences in the growth (Kunze 2001), resulting from internal wave induced mix- of the dispersion. Initially, there is a diffusive regime ing. Here, we show that surface-intensified processes in where dispersion increases linearly in time. This is fol- the mixed layer also generate significant lateral mixing lowed by a period in which the tracer experiences an in the interior. Current ML eddy parameterizations exponential growth in the dispersion because of the in- (Fox-Kemper et al. 2008) are designed to represent the tegrated action of strain. Finally, the fully developed effect of ML eddies within the surface ML. Our findings flow exhibits a Richardson-like power law in the growth suggest that such parameterizations may also need to of dispersion. The tracer variance is found to exhibit account for the effect of ML eddies in the pycnocline. these three regimes for a wide range of model parame-

Observational studies of mixing in the pycnocline have ters (H and by) and isopycnal surfaces on which dye is used dye dispersion (Ledwell et al. 1998; Sundermeyer released. The time-varying lateral diffusivity (Ky) con- and Ledwell 2001) to quantify lateral mixing in terms of sequently reflects the three dynamical regimes and a a horizontal Fickian diffusivity. But in our numerical dependence on the submesoscale parameters, depth of NOVEMBER 2011 B A D I N E T A L . 2099 isopycnal, and background stratification in the pycno- flat, so that initially no mean flow penetrates into the cline. Submesoscale filaments that are characterized by stratified region underlying the ML. The underlying the coincidence of large horizontal strain rate and vertical pycnocline has been initialized from a typical density vorticity act to rapidly flux the tracer and increase tracer profile from the North Atlantic taken from the World dispersion. However, the Nakamura effective diffusivity Ocean Atlas (Levitus 1982). Initially, the pycnocline is is small in the core of the filaments. The horizontal shear flat, with nonuniform stratification with an initial maxi- in the filaments overwhelms cross-frontal diffusion and mum value of N2 5 1.8 3 1024 s22 at 240 m and de- filament centers act as barriers to transport, while eddy creasing to N2 5 1025 s22 at 500 m, which is in general fluxes are enhanced only at the filament edges where agreement with open-ocean values (e.g., Lewis et al. tracer gradients are largest. 1986; Planas et al. 1999). For comparison, some of the numerical simulations of the upwelling fronts generated Acknowledgments. The authors thank Prof. Jean-Luc in the California Current system by Capet et al. (2008b),

Thiffeault for helpful discussions. Constructive com- their Figs. 5 and 10, have a slightly larger value of by ’ ments from two anonymous referees strengthened the 21.3 3 1027 s22 and smaller initial ML depth H ’ 50 m. manuscript. The authors are thankful to ONR (Awards The model is initialized with different values of lateral

N00014-09-1-0179 and N00014-09-1-0196) for support. buoyancy gradient 0.2by0,0.5by0,0.75by0,and5by0.The This study is a part of the ONR Departmental Research numerical integration with reference cross-front buoy-

Initiative ‘‘Scalable Lateral Mixing and Coherent Tur- ancy gradient of by0 is also integrated with an initial ML bulence.’’ depth H 5 100 and 150 m. Changes in ML depth result in changes to N2 at the base of the ML, as the ML base APPENDIX matches the pycnocline at different depths corresponding to different buoyancy frequencies. All the numerical ex- periments performed are summarized in Table 2. The Model Description model has fixed values for the diffusivity and the viscosity. 2 21 The model is an extension of the three-dimensional, In the horizontal, nh 5 5m s . In the vertical, nz 5 nonhydrostatic, free-surface model described in 1025 m2 s21. To study the time evolution of the tracer Mahadevan et al. (1996a,b, 2010). Model parameters are concentration variance along the pycnocline, east–west- summarized in Table 1. The model is set up in a zonally oriented streaks of tracer are initialized along different periodic channel with impermeable vertical walls at the isopycnals, namely at s 5 25.44, 26.19, 26.54, and 26.76 meridional boundaries and at the bottom. The hori- across the zonal extent of the domain. The streaks, of zontal dimensions are Ly 5 192 km in the meridional 1-km width and initial tracer concentration 1 unit, are direction, Lx 5 96 km in the zonal direction, and Htot 5 placed at y 5 80 km from the southern boundary (i.e., 500-m depth. The horizontal resolution is 1 km. In the below the surface ML front). To estimate effective dif- vertical the model has 32 layers, but the vertical reso- fusivity, a tracer with concentration varying linearly from lution is variable with depth, with enhanced resolution 1 at the southern boundary to 0 at the northern boundary of 2 m near the sea surface, decreasing to the resolution and spread along the entire domain is initialized along the of 36 m at depth. Lateral free-slip boundary conditions s 5 25.44, 26.19, 26.54, and 26.76 isopycnals. For the are applied. The model is nonhydrostatic, even though reference integration, the four isopycnal surfaces have the instabilities described are hydrostatic (Mahadevan initial depths of 216.59, 256.88, 302.72, and 354.87 m, re- 2006). The domain is initialized with a sharp lateral spectively. At the northern and southern boundary, the north–south density gradient in the upper layers, with tracer is subject to the boundary condition ›C/›y 5 0. isopycnals initially vertical. The southern half of the domain has lighter water, and is initialized using an an- REFERENCES alytic profile of the form Dr 5Dr tanh[0.03p(y 2 Ly/2)], which describes a surface ML front centered at 80 km Abernathey, R., J. Marshall, M. Mazloff, and E. Shuckburgh, 2010: Enhancement of mesoscale eddy stirring at steering levels in north of the southern boundary. In the reference in- the Southern Ocean. J. Phys. Oceanogr., 40, 170–184. tegration, the front has an initial cross-front density Batchelor, G. K., 1950: The application of the similarity theory of 23 contrast of Dr0 5 0.2 kg m and an initial cross-front turbulence to atmospheric diffusion. Quart. J. Roy. Meteor. 27 22 Soc., 133–146. buoyancy gradient of by0 520.9 3 10 s . The initial 76, depth of the ML is H 5 200 m and the initial ML Bennett, A. F., 1984: Relative dispersion: Local and nonlocal dy- 2 26 22 namics. J. Atmos. Sci., 41, 1881–1886. buoyancy frequency is N 5 10 s . The ML front is in Billant, P., and J.-M. Chomaz, 2000a: Experimental evidence for thermal wind balance, with the level of no motion set at a new instability of a vertical columnar vortex pair in a strongly the top of pycnocline where the isopycnals are initially stratified fluid. J. Fluid Mech., 418, 167–188. 2100 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41

——, and ——, 2000b: Theoretical analysis of the zigzag instability Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric front- of a vertical columnar vortex pair in a strongly stratified fluid. ogenesis models: Mathematical formulation and solution. J. Fluid Mech., 419, 29–63. J. Atmos. Sci., 29, 11–37. Bishop, C. H., 1993: On the behavior of baroclinic waves un- Houghton, R. W., 1997: Lagrangian flow at the foot of a shelfbreak dergoing horizontal deformation. I The ‘RT’ phase diagram. front using a dye tracer injected into the bottom boundary Quart. J. Roy. Meteor. Soc., 119, 221–240. layer. Geophys. Res. Lett., 24, 2035–2038. Boccaletti, G., R. Ferrari, and B. Fox-Kemper, 2007: Mixed layer Kamykowski, D., and S.-J. Zentara, 1986: Predicting plant nu- instabilities and restratification. J. Phys. Oceanogr., 37, 2228– trient concentrations from temperature and sigma-t in the 2250. upper kilometer of the world ocean. Deep-Sea Res., 33A, Capet,X.,J.C.McWilliams,M.J.Molemaker,andA.F.Shchepetkin, 89–105. 2008a: Mesoscale to submesoscale transition in the California Koszalka, I., J. H. LaCasce, and K. A. Orvik, 2009: Relative dis- Current system. Part I: Flow structure, eddy flux, and obser- persion in the Nordic Seas. J. Mar. Res., 67, 411–433. vational tests. J. Phys. Oceanogr., 38, 29–43. Kraichnan, R. H., 1967: Inertial ranges of two-dimensional turbu- ——, ——, ——, and ——, 2008b: Mesoscale to submesoscale lence. Phys. Fluids, 10, 1417–1423. transition in the California Current system. Part II: Frontal ——, and D. Montgomery, 1980: Two-dimensional turbulence. processes. J. Phys. Oceanogr., 38, 44–64. Rep. Prog. Phys., 43, 547–619. ——, ——, ——, and ——, 2008c: Mesoscale to submesoscale Kunze, E., 2001: Waves: Vortical mode. Encyclopedia of Ocean transition in the California Current system. Part III: Energy Sciences, S. T. J. Steele and K. Turekian, Eds., Academic balance and flux. J. Phys. Oceanogr., 38, 2256–2269. Press, 3174–3178. Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33–52. LaCasce, J. H., 2008: Statistics from Lagrangian observations. Einstein, A., 1905: Ueber die von der molekularkinetischen Theorie Prog. Oceanogr., 77, 1–29. der Waerme geforderte Bewegung von in ruhenden Fluessig- ——, and A. Bower, 2000: Relative dispersion in the subsurface keiten suspendierten Teilchen. Ann. Phys., 322, 549–560. North Atlantic. J. Mar. Res., 58, 863–894. Eldevik, T., and K. B. Dysthe, 2002: Spiral eddies. J. Phys. Ocean- ——, and A. Mahadevan, 2006: Estimating subsurface horizontal ogr., 32, 851–869. and vertical velocities from sea-surface temperature. J. Mar. Er-El, J., and R. L. Peskin, 1981: Relative diffusion of constant- Res., 64, 695–721. level balloons in the Southern Hemisphere. J. Atmos. Sci., 38, Lacorata, G., E. Aurell, B. Legras, and A. Vulpiani, 2004: Evidence 2264–2274. for a k25/3 spectrum from the EOLE Lagrangian balloons in Ferrari, R., and M. Nikurashin, 2010: Suppression of eddy mixing the low stratosphere. J. Atmos. Sci., 61, 2936–2942. across jets in the Southern Ocean. J. Phys. Oceanogr., 40, Ledwell, J. R., A. J. Watson, and C. S. Law, 1998: Mixing of a tracer 1501–1519. in the pycnocline. J. Geophys. Res., 103, 21 499–21 529. Fjørtoft, R., 1950: Application of integral theorems in deriving ——, T. F. Duda, M. A. Sundermeyer, and H. E. Seim, 2004: criteria of stability for laminar flows and for the baroclinic Mixing in a coastal environment: 1. A view from dye dispersion. circular vortex. Geofys. Publ., 17, 5–32. J. Geophys. Res., 109, C10013, doi:10.1029/2003JC002194. Fox-Kemper, B., R. Ferrari, and R. Hallberg, 2008: Parameteri- Levitus, S., 1982: Climatological Atlas of the World Ocean. NOAA zation of mixed layer eddies. Part I: Theory and diagnosis. Prof. Paper 13, 173 pp and 17 microfiche. J. Phys. Oceanogr., 38, 1145–1165. Levy, M., P. Klein, and A.-M. Treguier, 2001: Impacts of sub- Garrett, C., 1983: On the initial streakiness of a dispersing tracer in mesoscale physics on production and subduction. two- and three-dimensional turbulence. Dyn. Atmos. , J. Mar. Res., 59, 535–565. 7, 265–277. Lewis, M. R., W. G. Harrison, N. S. Oakey, D. Hebert, and T. Platt, Haine, T. W. N., and J. Marshall, 1998: Gravitational, symmetric, 1986: Vertical nitrate fluxes in the oligotrophic ocean. Science, and baroclinic instability of the ocean mixed layer. J. Phys. 234, 870–873. Oceanogr., 28, 634–658. Lin, J.-T., 1972: Relative dispersion in the enstrophy-cascading Haynes, P. H., 2001: Vertical shear plus horizontal stretching as inertial range of homogeneous two-dimensional turbulence. a route to mixing. Proc. 12th ‘Aha Huliko’a Hawaiian Winter J. Atmos. Sci., 29, 394–395. Workshop, Manoa, HI, University of Hawaii, 70–73. Lumpkin, R., and S. Elipot, 2010: Surface drifter pair spreading in ——, and J. Anglade, 1997: The vertical-scale cascade in atmo- the North Atlantic. J. Geophys. Res., 115, C12017, doi:10.1029/ spheric tracers due to large-scale differential advection. 2010JC006338. J. Atmos. Sci., 54, 1121–1136. Mahadevan, A., 2006: Modeling vertical motion at ocean fronts: ——, and E. Shuckburgh, 2000a: Effective diffusivity as a di- Are nonhydrostatic effects relevant at submesoscales? Ocean agnostic of atmospheric transport 1. Stratosphere. J. Geophys. Modell., 14, 222–240. Res., 105, 22 777–22 794. ——, and A. Tandon, 2006: An analysis of mechanisms for sub- ——, and ——, 2000b: Effective diffusivity as a diagnostic of at- mesoscale vertical motion at ocean fronts. Ocean Modell., 14, mospheric transport 2. Troposphere and lower stratosphere. 241–256. J. Geophys. Res., 105, 22 795–22 810. ——, J. Oliger, and R. L. Street, 1996a: A nonhydrostatic meso- Held, I. M., R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson, scale ocean model. Part I: Well-posedness and scaling. J. Phys. 1995: Surface quasi-geostrophic dynamics. J. Fluid Mech., 282, Oceanogr., 26, 1868–1880. 1–20. ——, ——, and ——, 1996b: A nonhydrostatic mesoscale ocean Herterich, K., and K. Hasselmann, 1982: The horizontal diffusion model. Part II: Numerical implementation. J. Phys. Oceanogr., of tracers by surface waves. J. Phys. Oceanogr., 12, 704–711. 26, 1881–1900. Holmes-Cerfon, M., O. Buehler, and R. Ferrari, 2011: Particle ——, A. Tandon, and R. Ferrari, 2010: Rapid changes in mixed layer dispersion by random waves in the rotating Boussinesq sys- stratification driven by submesoscale instabilities and winds. tem. J. Fluid Mech., 670, 150–175. J. Geophys. Res., 115, C03017, doi:10.1029/2008JC005203. NOVEMBER 2011 B A D I N E T A L . 2101

Marshall, J., E. Shuckburgh, H. Jones, and C. Hill, 2006: Estimates Smith, K. S., and G. K. Vallis, 2001: The scales and equilibration of and implications of surface eddy diffusivity in the Southern midocean eddies: Freely evolving flow. J. Phys. Oceanogr., 31, Ocean derived from tracer transport. J. Phys. Oceanogr., 36, 554–571. 1806–1821. ——, and R. Ferrari, 2009: The production and dissipation of McWilliams, J. C., F. Colas, and M. J. Molemaker, 2009a: Cold compensated thermohaline variance by mesoscale stirring. filamentary intensification and oceanic surface convergence lines. J. Phys. Oceanogr., 39, 2477–2501. Geophys. Res. Lett., 36, L18602, doi:10.1029/2009GL039402. ——, and J. Marshall, 2009: Evidence for enhanced eddy mixing at ——, M. J. Molemaker, and E. I. Olafsdottir, 2009b: Linear fluc- middepth in the Southern Ocean. J. Phys. Oceanogr., 39, 50–69. tuation growth during frontogenesis. J. Phys. Oceanogr., 39, Spall, M., 1997: Baroclinic jets in confluent flows. J. Phys. Ocean- 3111–3129. ogr., 27, 381–402. Molemaker, M. K., J. C. McWilliams, and I. Yavneh, 2005: Baro- Stommel, H., 1949: Horizontal diffusion due to oceanic turbulence. clinic instability and loss of balance. J. Phys. Oceanogr., 35, J. Mar. Res., 8, 199–225. 1505–1517. Stone, P., 1966: On non-geostrophic baroclinic stability. J. Atmos. Morel, P., and M. Larcheveque, 1974: Relative dispersion of constant- Sci., 23, 390–400. level balloons in the 200-mb general circulation. J. Atmos. Sci., ——, 1970: On non-geostrophic baroclinic stability: Part II. J. Atmos. 31, 2189–2196. Sci., 27, 721–726. Munk, W., 1981: Internal waves and small-scale processes. Evolu- ——, 1971: Baroclinic stability under non-hydrostatic conditions. tion of Physical Oceanography, B. Warren and C. Wunsch, J. Fluid Mech., 46, 659–671. Eds., MIT Press, 264–291. Sundermeyer, M. A., and J. F. Price, 1998: Lateral mixing and the Nagai, T., A. Tandon, and D. L. Rudnick, 2006: Two-dimensional North Atlantic Tracer Release Experiment: Observations and ageostrophic secondary circulation at ocean fronts due to numerical simulations of Lagrangian particles and passive vertical mixing and large-scale deformation. J. Geophys. Res., tracer. J. Geophys. Res., 103, 21 481–21 497. 111, C09038, doi:10.1029/2005JC002964. ——, and J. R. Ledwell, 2001: Lateral dispersion over the conti- Nakamura, N., 1996: Two-dimensional mixing, edge formation, nental shelf: Analysis of dye-release experiments. J. Geophys. and permeability diagnosed in an area coordinate. J. Atmos. Res., 106, 9603–9622. Sci., 53, 1524–1537. ——, and M.-P. Lelong, 2005: Numerical simulations of lateral ——, and J. Ma, 1997: Modified Lagrangian-mean diagnostics of dispersion by the relaxation of diapycnal mixing events. the stratospheric polar vortices 2. Nitrous oxide and seasonal J. Phys. Oceanogr., 35, 2368–2386. barrier migration in the cryogenic limb array etalon spec- ——, J. R. Ledwell, N. S. Oakey, and B. J. W. Greenan, 2005: trometer and SKYHI general circulation model. J. Geophys. Stirring by small-scale vortices caused by patchy mixing. Res., 102, 25 721–25 735. J. Phys. Oceanogr., 35, 1245–1262. Okubo, A., 1970: Horizontal dispersion of floatable particles in the Tandon, A., and C. Garrett, 1994: Mixed layer restratification due to vicinity of velocity singularity such as convergences. Deep-Sea a horizontal density gradient. J. Phys. Oceanogr., 24, 1419–1424. Res., 17, 445–454. ——, and ——, 1995: Geostrophic adjustment and restratification ——, 1971: Oceanic diffusion diagrams. Deep-Sea Res., 18, 789–802. of a mixed layer with horizontal gradients above a stratified Ollitrault, M., C. Gabillet, and A. C. de Verdiere, 2005: Open ocean layer. J. Phys. Oceanogr., 25, 2229–2241. regimes of relative dispersion. J. Fluid Mech., 533, 381–407. Taylor, G. I., 1921: Diffusion by continuous movements. Proc. Roy. Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed. Springer, Soc. London, 20, 196–211. 728 pp. ——, 1953: Dispersion of soluble matter in solvent flowing slowly Planas, D., S. Agusti, C. M. Duarte, T. C. Granata, and M. Merino, through a tube. Proc. Roy. Soc. London, 219, 186–203. 1999: Nitrate uptake and diffusive nitrate supply in the Central Thiffeault, J.-L., 2008: Scalar decay in chaotic mixing. Transport Atlantic. Limnol. Oceanogr., 44, 116–126. and Mixing in Geophysical Flows, J. B. Weiss and A. Pro- Polzin, K. L., and R. Ferrari, 2004: Lateral dispersion in NATRE. venzale, Eds., Lecture Notes in Physics, Vol. 744, Springer, J. Phys. Oceanogr., 34, 247–257. 3–35. Richardson, L. F., 1926: Atmospheric diffusion shown on a distance- Thomas, L., and C. M. Lee, 2005: Intensification of ocean fronts by neighbour graph. Proc. Roy. Soc. London, 110, 709–737. down-front winds. J. Phys. Oceanogr., 35, 1086–1102. Rudnick, D. L., and R. Ferrari, 1999: Compensation of horizontal Weiss, J., 1991: The dynamics of enstrophy transfer in two- temperature and salinity gradients in the ocean mixed layer. dimensional hydrodynamics. Physica D, 48, 273–294. Science, 283, 526–529. Winters, K. B., and E. A. D’Asaro, 1996: Diascalar flux and the rate Shuckburgh, E., and P. Haynes, 2003: Diagnosing transport and of fluid mixing. J. Fluid Mech., 317, 179–193. mixing using a tracer-based coordinate system. Phys. Fluids, Young, W. R., 1994: The subinertial mixed layer approximation. 15, 3342–3357. J. Phys. Oceanogr., 24, 1812–1826. ——, H. Jones, J. Marshall, and C. Hill, 2009: Robustness of an ——, P. B. Rhines, and C. J. Garrett, 1982: Shear-flow dispersion, effective diffusivity diagnostic in oceanic flows. J. Phys. Ocean- internal waves and horizontal mixing in the ocean. J. Phys. ogr., 39, 1993–2009. Oceanogr., 12, 515–527.